Regular maps and principal congruence subgroups of Hecke groups

Regular q-valent maps correspond to normal subgroups of the triangle group (2, q, ∞). This group has a representation as the Hecke group Hq which is generated by z → -1/z and z → -1/z+λq, where λq := 2 cos π/q. We investigate the regular maps corresponding to the principal congruence subgroups of Hq. Those of low index give many interesting regular maps.